1. Shell the peas, collecting them in a bowl (you should have about 2 cups) and putting the pods in a large saucepan. Rinse the pods under running water, and then add the onion and 8 cups of water, the salt and the sugar. Bring to a simmer and cook until the pods are fairly soft and quite fragrant, about 15 minutes. Strain into a measuring cup, discarding the cooked pods. You should have about 6 cups of broth.

2. Rinse out the saucepan and heat the butter and leek in it. Cook over medium-low heat, stirring, until the leek softens, about 5 minutes. Add the rice and stir to coat well. Add 5 cups of the broth and bring to a simmer. Cook until the rice is quite soft, about 20 minutes. The kernels should be soft enough that they're beginning to split at the ends and should offer very little resistance when tasted.

3. Add the peas and increase the heat to high. Cook just until their color brightens and they soften, 3 to 5 minutes.

4. Ladle about half of the brothy pea and rice mixture into a blender. Cover the top with a towel to prevent any splashing and pulse once or twice on low speed to get the mixture started. Once the mixture has been chopped, blend at low speed about 30 seconds, and then raise the speed to high for about 10 seconds, until the mixture is a smooth purée.

5. Strain the mixture into a pot, stirring it and rubbing it against the bottom and sides of the strainer with a rubber spatula until all that is left is a fairly dry mixture of broken rice and pea skins to be discarded. Repeat with the remaining soup. You should have about 3 cups of soup; if necessary, add a little more of the broth to make that amount. Taste and adjust the seasoning. Refrigerate, tightly covered until chilled. (The recipe can be prepared to this point up to 4 hours in advance.)

6. When ready to serve, gently stir together the crab and the mint in a small bowl, being careful not to break up the crab meat. Whisk lemon juice into the pea soup and taste and add more salt or lemon if necessary. Ladle about three-fourths cup of soup into each of four shallow soup plates or pasta bowls. Carefully spoon about 2 tablespoons of the crab mixture into the center of the soup and serve.